The significance of chemical reaction, thermal buoyancy, and external heat source to optimization of heat transfer across the dynamics of Maxwell nanofluid via stretched surface

Energy loss during the transportation of energy is the main concern of researchers and industrialists. The primary cause of heat exchange gadget inefficiency during transportation was applied to traditional fluids with weak heat transfer characteristics. Instead, thermal devices worked much better when the fluids were changed to nanofluids that had good thermal transfer properties. A diverse range of nanoparticles were implemented on account of their elevated thermal conductivity. This research addresses the significance of MHD Maxwell nanofluid for heat transfer flow. The flow model comprised continuity, momentum, energy transport, and concentration equations in the form of PDEs. The developed model was converted into ODEs by using workable similarities. Numerical simulations in the MATLAB environment were employed to find the outcomes of velocity, thermal transportation, and concentration profiles. The effects of many parameters, such as Hartman, Deborah, buoyancy, the intensity of an external heat source, chemical reactions, and many others, were also evaluated. The presence of nanoparticles enhances temperature conduction. Also, the findings are compared with previously published research. In addition, the Nusselt number and skin friction increase as the variables associated with the Hartman number and buoyancy parameter grow. The respective transfer rates of heat are 28.26\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\%$$\end{document}% and 38.19\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\%$$\end{document}% respectively. As a result, the rate of heat transmission increased by 14.23\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\%$$\end{document}%. The velocity profiles enhanced while temperature profiles declined for higher values of the Maxwell fluid parameter. As the external heat source increases, the temperature profile rises. Conversely, buoyancy parameters increase as it descends. This type of problem is applicable in many fields such as heat exchangers, cooling of electronic devices, and automotive cooling systems.

In recent decades, there has been significant interest in the production and application of microdevices.Microtechnology mechanisms offer numerous benefits, including fabricating microdevices (e.g., microsensors, microvalves, and micropumps) in minute dimensions and with significant efficiency.The magnification of efficiency and keeping them cool during work is an apprehension.There has been tremendous research on the flow and transfer of nanofluids, and it is still an active research area.Nanofluids play a very important role in heat and mass transfer phenomena. 1is one of the pioneers of this field who introduced nanoparticles in fluid flow after remarkable changes happened in heat mass transformation and fluid flow.Then 2 investigated the Tiwari-Das model with the Dufour Sorrot effect over Al 3 -water nanofluid flow over the needle. 3worked the magnetic field, thermal radiation, and convection effects in a square body filled with TiO 2 . 4,5discussed the stagnation point flow for the Tiwari and Das model over a stretching sheet with a slip effect. 6investigated the impact of MHD effects shape of the nanoparticles and thermal radiations through the sheet.
There are many applications of magnetic hydrodynamics, including liquid flow and temperature transport, peristaltic flow, ship propulsion, metallurgy, crystal growth, fusion reactors, etc.Researchers use magnetic nanoparticles.At present, there are a variety of industries that use magnetic fields.The use of MHD nanofluids in biological imaging and several other fields is wide-ranging. 7,8used the FEM method to look at the flow of MHD nanofluid and see how melting affected Cattaneo-Christov and thermal radiation. 9discussed the MHD nanofluid flow between two cylinders in a theoretical analysis.The results showed that while the temperature profile decreased with rising Hartmann numbers and radiation criteria, it improved with the ascent of Reynolds and Eckert numbers.An octagon container with fins was used in the work by 10 to analyse the entropy of buoyancydriven magnetohydrodynamic hybrid nanofluid flow.The results of the investigation showed that while the magnetic number decreased with nanoparticle concentrations, entropy rose. 11discussed ferrofluid movement across a rotating disc in the presence of a strongly fluctuating magnetic field.The findings showed that a higher field frequency raised the temperature, while a smaller nanoparticle diameter decreased heat transmission.The stability analysis of an MHD hybrid nanofluid flow with a quadratic speed over a stretching sheet was talked about in 12 .The initial solution's positive minimum eigenvalue was shown by stability analysis, which adequately defined a steady and achievable flow. 13investigated the MHD fluid flow with multi-slip effects over the permeable sheet.The thermodynamic properties of hybrid and traditional nanofluid flows across a curvy, sliding porous surface were studied by 14 .Additionally, it was believed that the surface is wrapped inside a circular sphere.Many other researchers also work on MHD fluid flow, like [15][16][17][18] .
These days, fluid flow under the influence of mixed convection appeals to researchers.Mixed convection phenomena are widely employed in the construction of industrial techniques, e.g., cooling of electronic devices, heat exchanging from nuclear reactors, biomedical sciences, and many other technological applications. 19investigated heat transfer through the flow of a nanofluid in the presence of a magnetic effect through a rotating system.Because of the Lorentz forces, it was discovered that the Nusselt number fell as the magnetic parameter increased. 20,? elaborated on the nanofluid Brownian motion and thermophoresis through a rotating stretching sheet. 21explored the fluctuation in temperature profiles of MHD nanofluid flow over the stretching velocity of CU-water over the rotating frame.Enhancing the combined convection and magnetic parameters produces a delay in the boundary layer separation. 22discussed the bouncy and effects of radiation for MHD micropolar nanofluid flow over rotating pours stretching sheet a numerical investigation.The velocity profile decreased as the magnetic parameter M increased.Conversely, when the magnetic parameter M grew, so did the micro-rotation.The movement of a MWCNT-Fe3O4-water hybrid nanofluid across a micro-wavy conduit was investigated by 23 .
The stretching/shrinking phenomena are one of the key characteristics of nanofluid flow.The pioneer of this study is 24 , who introduced the phenomena.This concept was later used by many scientists to achieve further accomplishments like Hayat et al. 25 discussed the heat transportation with stretchable sheets for MHD flow. 26xamined the relevance of temperature gradient-induced fluctuation in the diameter, mass flow, and dissipation of energy in the fluid's convection radiation motion, the heat source, and the Darcy-Forchheimer model across a cylinder.Because of the density spectrum, the energy fluxion was noticeably larger in magnitude due to an enlarged nanoparticle's diameter, which also dramatically reduced the temperature profile across the domain. 27

Purposed model
A two-dimensional laminar Maxwell nanofluid flow with steady boundary conditions was evaluated under the influence of activation energy, mixed convection, and stretching effects, about Fig. 1.A constant magnetic field is applied perpendicularly to the flow surfaces to balance the boundary layers with a permanent heat source.Consider that u w (x) = ax, a > 0 , and u e (x) = bx, b > 0 are the stretching velocity and stream velocity respec- tively.Now the governing equations for fluid flow, heat transportation, and consecration of nanoparticles are given below [30][31][32] (1) Here the velocity along the x-axis is u and the y-axis is v respectively.In the above equations, a few terms appeared e.g.B o magnetic field, σ * electric conductivity, the thermal expansion is β t , the density of fluid is ρ , T is the ambient temperature, Q is heat source coefficient, the concentration expansion is In order to advance the investigation, We'll apply the similarity transformation below.Considering 30 .
The Eq. ( 1) is satisfied identically by employing transformation, from Eqs. (2-4) converted into nonlinear ordinary differential equations as The boundary conditions also converted In the proposed model, which is mentioned in equations 8-10, a few dimensionless values appear after leveraging the similarities mentioned above and simplifying the above model.Here, M = σ B 2 ρb known as the Hartman number, P r = ν α , known as the Prandtl number, Nr = Gr Re 2 known as the free convection parameter, Gr = is the Grashof number, R e = ux ν is Reynolds number 33 is thermal free convection parameter, Q s = Q ρbC p is heat generation parameter, and and is known as the value of thermophoresis.The skin friction coefficient and Nusselt number are defined as By using the similarity transformations from Eq. ( 7) into Eqs.( 13) and ( 14), we obtain (2) ( Vol.:(0123456789)

Numerical scheme
The solution of nonlinear ODEs (8-10) has been required to find the solution of appeared physical quantities in the proposed model.The Runnga-kutta numerical method is applied to reduce the nonlinearity of the model.The solution has been found with the help of MATLAB.The higher order ODEs converted as follows.
The boundary conditions are indeed modified similarly.

Model validation
We drew a comparison with the existing litterateur for validation of the proposed flow model.The most significant results have been compared in Tables 1 and 2 to previous research of 30,34 , and the validity of the results is proved for limited cases.The mathematical model uses the table and fixed, dimensionless numbers for parameters to look at what happens when thermal radiation, viscosity, and Arrhenius energy change.We h a v e β = 1.0, = 0.15, M = 1.0,Pr = 2.0, Sc = 1.0, σ = 0.50, δ = 0.3, m = 0.5, E = 0.3, Nb = 0.15, Nt = 0.21, Nr = 0.2, andRb = 0.2 .These comparison tables show the validation of the results with existing literature comprehensively.

Results and discussion
In this chapter, we investigated the magnetized Maxwell Nanofluid flow with activation energy, mixed convection, and heat source over the stretching sheet under the limiting condition with flowing fixed values: .

Pr 34 30
New results New results www.nature.com/scientificreports/0.2, Q s = 0.2, ǫ = 0.2 .The impact of these parameters is discussed over the momentum profile, energy profile, and concentration profile of nanofluid.In Figs. 2, 3 and 4 the behavior of the velocity profile is discussed for different parameters.The velocity profile shows dynamically increasing behavior due to the increased values of Harmat number M, Mexwell parameter β , stretching parameterǫ , buoyancy parameter , and density ratio Nr respectively.An increase in Hartmann Number in Fig. 2 can reduce the fluid's boundary layer movement.Since the Lorentz force is created by the transversal magnetic field in electrically conductive fluids.Free stream speed, however, transcends the stretching surface speed when the magnetic parameter is countered with the velocity distribution 35,36 .When the Maxwell parameter β (Deborah number) as shown in Fig. 2 increases, it indicates that elastic response dominates viscous response.Hence, the fluid can store and release elastic energy significantly.Due to its elasticity, the fluid is capable of accelerating and increasing its velocity more rapidly when subjected to external forces or disturbances.Figure 3 shows the impact of the density ratio Nr on the velocity profile.The velocity profile increases as the  www.nature.com/scientificreports/density ratio increases, indicating that the density ratio aids in the system's stabilization.When the buoyancy parameter through Fig. 3 increases, a difference in densities within the fluid and outside the fluid will take place fluid gets heated and becomes less dense, it tends to rise, while the cooler.As a result, the buoyancy forces become more pronounced and can drive more vigorous fluid motion.Thus, the fluid accelerates, and its velocity increases.When the stretching parameter ǫ increases the cross-sectional area of the flow geometry decreases, causing the fluid to accelerate to maintain a constant flow rate as indicated in Fig. 4. Thus, the fluid accelerates, and its velocity increases.In Figs.4b, 5, 6 and 7 the behavior of the temperature profile is discussed for different parameters.The temperature profile shows dynamically increasing behavior for the boosted values of Brownian motion Nb, thermophoresis Nt, and heat sources Qs but for maxwell parameter β , Harmat number, stretching parameter ǫ , and buoyancy parameter showed diminished behavior respectively.When the stretching parameter ǫ is increased, the internal energy of the fluid is converted into work during the stretching process as shown in Fig. 4. The volume www.nature.com/scientificreports/ of the fluid grows while the pressure falls as it is stretched.The ideal gas law states that as a gas's pressure lowers, so does its temperature.The temperature gradient in the fluid is raised as the thermophoresis Nt is increased as depicted in Fig. 6, creating a force that acts on the suspended particles.This force can cause particles to travel towards regions of higher temperature.In Fig. 8 , the actions that were taken on the concentration profile are discussed for different parameters.The concentration profile shows dynamically increasing behavior for the boosted values of activation energy E but reaction rate σ showed diminished behavior.
In Figs. 9 and 10, the behavior of skin friction and Nusselt number are discussed for different parameters.The skin friction and Nusselt number show dynamically increasing behavior due to the increased values of Harman number M, and respectively.The presence of M and can cause the fluid to mix more vigorously, leading to improved heat transfer.This enhanced mixing helps in breaking up boundary layers and increasing the convective heat transfer coefficient, thus increasing the Nusselt number.The increase in skin friction due to the Hartman number and can cause flow distortion and energy losses in the system.This effect is particularly significant in situations where minimizing energy losses and optimizing flow efficiency are crucial, such as in certain industrial applications, energy generation systems, or aerospace engineering.

Conclusions
This research aims to explore how magnetized nanofluid and Maxwell fluid flow under the influence of mixed convection with convection and heat source over two-dimensional stretching sheets with activation energy.The Rungaa Kutta method is used to find the mathematical results for the velocity profile, temperature profile, concentration profile, skin friction coefficient, and Nusselt number of the above-mentioned problems.Below is a summary of a few of the most important results.
• The velocity profile f ′ (η) is increasing noticeably with the boosted values of parameters magnetic parameter, Maxwell fluid parameter, thermal mixed convection parameter, velocity ratio parameter, and concentration mixed convection parameter.• The temperature profile θ(η) is going up when the parameters thermophoresis, Brownian motion, and an external heat source are raised.On the other hand, it goes down when the Maxwell fluid parameter, stretching parameter, Hartman number, and buoyancy parameter are raised.• Nanoparticle concentration profiles get lower as Brownian motion and chemical reaction parameters go up, and they change shape as boosted values of thermophoresis and activation energy go up.• The skin friction coefficient and Nusselt number show increasing behavior for increasing valves of Hartman number and buoyancy parameter.

Future directions
This work serves as a foundation for further exploration in the emerging field of MHD bio-convective threedimensional rotating flow of nanofluids across stretched surfaces.Possible areas for additional investigation may include: • Interdisciplinary collaboration: Foster interdisciplinary collaboration between researchers in fluid dynam- ics, bioengineering, and materials science.This collaboration can lead to innovative solutions and a more holistic understanding of the complex interactions within these systems.• Exploration of novel materials: Investigate the use of novel nanomaterials with specific properties to enhance the thermal conductivity and bio-convection effects.This can open avenues for the development of advanced nanofluids with tailored characteristics for specific applications.
, ǫ = a b indicates velocity ratio parameter, Nb = τ D b (C w −C ∞ ) ν known as the value of Brownian motion, β = b * denotes Maxwell fluid parameter,

Figure 2 .
Figure 2. Behaviour of velocity under the influence of M and β.

Figure 3 .Figure 4 .
Figure 3. Behaviour of velocity under the influence of Nr and .

Figure 5 .Figure 6 . 6 Figure 7 .
Figure 5. Behavior of temperature under the influence of and M.

Figure 8 . 3 Figure 9 . 3 Figure 10 .
Figure 8. Behaviour of concentration under the influence of E and δ.